3 Answers
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If you examine the circle of fifths, this will help visualize why the orders of sharps and flats occur in their respective sequences. For sharps, we begin with C. The next item in the circle is the key of G, with one sharp in the key signature: F. The next key is D, with two sharps, F and C. Next comes the key of A, with F, C, and G in the key signature. Following this pattern, you wind up with the FCGDAEB pattern. C is a fifth up from F, G is a fifth up from C, and so on. Going the other direction (where we instead to refer to the same chart as the circle of fourths), we have a similar pattern, using fourths beginning with B and resulting in the BEADGCF pattern. E is a fourth up from B, A a fourth up from E, and so on.

Thanks, that explains it nicely. The circle of fifths explains a lot of things quite nicely, definitely printing it out!
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JesseDec 2 '11 at 18:32

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I never heard about the circle of fourths? For me it's still fifth, only going in the other direction. I like seeing it as a double circle, going from 7b (one o'clock) to 7# (11 o'clock), through zero (12 o'clock). Alternatively, showing all 15 in one circle, with overlap for B major / Cb major, F# major / Gb major, and so on.
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GauthierDec 6 '11 at 10:13

@Gauthier "Circle of fifths" is indeed the most common reference, and if read clockwise in sharps you do indeed ascend in fifths (FCGDAEB). Many refer to the counterclockwise reading of the same chart as the "circle of fourths," as then you're ascending in fourths (BEADGCF). But arguably and as you well stated, the same could be said of just descending in fifths rather than ascending in fourths. At any rate, simply referring to it as the circle of fifths does just fine in all contexts, in my opinion. ;)
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Joe LewisDec 7 '11 at 23:40

The order of sharps and flats is as it is because, in preserving the whole-step/half-step arrangement of the major and (natural) minor scales, the sharps and flats appear in that order. That is, there are no major keys with E♭ that do not also have B♭, but there is a major key (F Major) that has B♭ but still has E♮. The order of sharps works similarly.

It's purely academic.When a key has, say, 3#, it's in A maj. or F# min.and the three sharps will always be the same, i.e. F# C# G#.We used to play music written by someone who just scribbled the sharps anywhere! All we needed to do was count how many - they had to be the right ones for the key, and that's predetermined.